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%I #69 Oct 23 2023 06:54:33
%S 0,122,3113,719377,15467683
%N The maximum integer M such that there are no prime n-tuplets of any possible pattern between M^2 and (M+1)^2, or -1 if no such maximum M exists.
%C All terms are conjectural. A prime n-tuplet is defined as the densest permissible prime constellation containing n primes. The term a(2) corresponds to twin primes, a(3) to prime triplets, a(4) to prime quadruplets, etc. Extensive computational evidence suggests that these terms are valid. However, there is no proof that the greatest integer M exists - not even for a subset of values of n. If one could find a constructive existence proof, then Twin Prime Conjecture as well as Legendre's Conjecture would require just a trivial additional step. - Edited by _Hugo Pfoertner_, Sep 15 2021
%C Note that, for some n, a prime (n+1)-tuple must include a prime n-tuple; e.g., prime quadruplets include prime triples. Thus, if any term is -1, subsequent terms may be -1, too. - _Franklin T. Adams-Watters_ and _Alexei Kourbatov_, Jul 14 2011
%C However, for other n, a prime (n+1)-tuple does NOT include a prime n-tuple; e.g. 7-tuples {p, p + 2, p + 6, p + 8, p + 12, p + 18, p + 20} do not contain 6-tuples {p-4, p, p + 2, p + 6, p + 8, p + 12}; see List of all possible patterns of prime k-tuplets by Tony Forbes.
%C Assuming the Hardy-Littlewood k-tuple conjecture, the average distance between k-tuples grows slower than the distance between consecutive squares. This is an indication (but not a proof) that the maximum integer M in this sequence does exist for all n.
%C a(6) > 3005845357, because there is a gap of size 7191214380 between consecutive sextuplets, enclosing in its interior the interval between the two squares 3005845357^2 and 3005845358^2, found with a fast sieving program provided by _Martin Raab_, i.e., 9035106309825245467 < 3005845357^2 = 9035106310198457449 < 3005845358^2 = 9035106316210148164 < 9035106317016459847. Statistical considerations based on the observed scatter of maximum gap sizes (see A200503 and data provided at N. Luhn link) suggest a range of 3*10^9 < a(6) < 6*10^9. In a private communication _Martin Raab_ provided the following estimates for the order of magnitude of the next terms: a(7) ~= 1.1*10^11, a(8) ~= 1.6*10^12, a(9) ~= 6*10^13, a(10) ~= 3*10^16. - _Hugo Pfoertner_, Jul 31 2023, corrected Oct 23 2023
%H Tony Forbes and Norman Luhn, <a href="https://pzktupel.de/ktpatt_hl.php">Patterns of prime k-tuplets & the Hardy-Littlewood constants</a>
%H G. H. Hardy and J. E. Littlewood, <a href="https://dx.doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
%H A. Kourbatov, <a href="http://arxiv.org/abs/1301.2242">Maximal gaps between prime k-tuples: a statistical approach</a>, arXiv preprint arXiv:1301.2242, 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kourbatov/kourbatov3.html">J. Int. Seq. 16 (2013) #13.5.2</a>
%H Norman Luhn, <a href="https://pzktupel.de/RecordGaps/GAP06.php">Record Gaps Between Prime Sextuplets, up to 10^17</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>
%e The term a(4)=719377 means that there are no prime quadruplets between 719377^2 and 719378^2, but there are prime quadruplets between m^2 and (m+1)^2 for m > 719377.
%Y Cf. A091592: Numbers n such that there are no twin primes between n^2 and (n+1)^2; A008407: Minimal width of prime n-tuplet.
%Y Cf. A020497, A200503.
%K nonn,hard,more
%O 1,2
%A _Alexei Kourbatov_, Jul 11 2011
%E First term, 0, added and offset changed by _Zak Seidov_, Jul 11 2011
%E Clarification regarding patterns in the title added by _Hugo Pfoertner_, Sep 15 2021
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